The shapes of things to come.
Every mathematician gets excited finding a new breakthrough – it’s like a mountain climber finding a new route to climb in the Himalayas. Sometimes these breakthroughs have applications to things we know, but most of the time the applications are not apparent. We just enjoy answering mathematical questions for what they are and finding answers to these abstract theories.
It’s like solving a puzzle. It all starts with the desire to answer questions using analytical thinking and our problem-solving mind. This is why graduates in mathematics are sought after in a lot of industries, not necessarily based on maths: they’re willing to inquire and analyse to find solutions.
Some people can work things out in their head – they’re the super geniuses. But I need paper and pen – and I make a mess! The floor of my workspace is covered with papers. You are always going to stumble and make mistakes, sometimes massive mistakes! We are definitely not perfect. I often put mistakes in my lectures to see which students fish them out. It’s part of the process – it’s a fun thing.
Maths is hidden in every single thing in nature. If somebody can describe this movement or change, then you can study it.
Everyone assumes that I’m really good with numbers. I’m not! I do shapes. I’m a very visual person. My specialty is geometric analysis. I take a geometric object and try to describe its properties with a partial differential equation.
When I look at anything that happens in everyday life, I’m always, “How can that movement or shape be described?” You see a fire and how it burns a piece of paper – I’m like, “How can I describe that?” My brain just works this way. How is the wind moving those branches? How do the waves arrive on the shore? Maths is hidden in every single thing in nature. If somebody can describe this movement or change, then you can study it.
I’ve been focused lately on geometric objects that move based on an energy. That’s what nature does: it tries to minimise or maximise some sort of energy. So we take that energy and do something called calculus of variations – then we can obtain a Euler-Lagrange equation too. This way we get our equations that we can study.
Some of my work is now used to model how fire fronts merge. I’m not an expert in bushfires at all, but I know Australia has been plagued by them. I was talking to Jason Sharples from UNSW Canberra, who is Professor of Bushfire Dynamics; he’s an expert in modelling, with a background in pure mathematics and curvature flow. We were discussing how we can model moving fronts. When you simplify a bushfire, it’s basically just a line that you put in a plane. This “real” fire line has similarities in its movement to the curvature flows that we’re studying. We have some interesting ideas there that we are working on.
When you simplify a bushfire, it’s basically just a line that you put in a plane.
I’m also focusing at the moment on trying to answer questions of minimal surfaces and constantly curvatured surfaces with boundaries. Think about foams and bubbles – take two or three bubbles and cluster them together. How would they stay in equilibrium form? After a while the bubbles start popping. We’re thinking about where that configuration can become stable. And we’re trying to solve that using curvature flow. Then the question starts, what sort of minimal surface can you have? We’re now looking at double bubble conjectures and things like this.
I’ve also recently been working together with a couple of collaborators on clusters and space partition problems. We’re trying to think how we can partition the space in X numbers of regions that each contain a certain volume and then have a certain area, and we’re trying to do this using a curvature flow. These clusters or partition problems have applications when describing foams or other materials. You can use them, for example, when you want to construct new materials which are lightweight but have high strength, or to improve thermal or acoustic insulation.
We’re just the mathematicians who think: how can I do this? And then: what’s the next big problem?
If we’re able to create a complete theory about it, maybe then we can put it into practice by constructing foams which might be used in the oil or gas industry to recover oil spills or extinguish fires. Or it might be used in the food industry to stabilise things that you want to put on the shelf, like whipped cream. Or there might be direct application in biology, for example, if somebody is trying to construct some sort of bone material – that might also involve a cluster of cells.
The applications could eventually be found in many fields of industry, but that’s not our work. We’re just the mathematicians who think: how can I do this? And then: what’s the next big problem?
As told to Graem Sims for Cosmos Weekly
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Valentina Wheeler
Dr Valentina Wheeler is a Senior Research Fellow in the School of Mathematics and Applied Statistics at the University of Wollongong, and the winner of the 2023 Christopher Heyde Medal, awarded by the Australian Academy of Science for her contribution to mathematics.
